An Introduction to Lieb's Simplified approach to the Bose gas

TL;DR

The book develops Lieb's Simplified approach to the Bose gas, replacing the full many-body problem with nonlinear, nonlocal single-particle equations (Big, Medium, Simple), and derives a closure via a factorization assumption. The Simple Equation yields an energy per particle $e$ that matches Lee-Huang-Yang predictions in the low-density limit and the high-density limit for positive-type potentials, while the Big/Medium equations provide accurate descriptions across densities; condensate fractions and other observables align with Bogoliubov theory at low density and show promising behavior at intermediate densities. The work provides rigorous existence results for the Simple Equation, analyzes asymptotics for energy and condensate fraction, derives and proves the Complete Equation (and its reductions), and develops numerical tools to study the Big/Medium equations, while outlining open problems and potential extensions to broader settings. Overall, the approach offers a tractable, predictive framework that captures essential Bose gas physics across density regimes and suggests directions for linking to the full many-body gas and to other quantum systems.

Abstract

This is a book about Lieb's Simplified approach to the Bose gas, which is a family of effective single-particle equations to study the ground state of many-body systems of interacting Bosons. It was introduced by Lieb in 1963, and recently found to have some rather intriguing properties. One of the equations of the approach, called the Simple equation, has been proved to make a prediction for the ground state energy that is asymptotically accurate both in the low- and the high-density regimes. Its predictions for the condensate fraction, two-point correlation function, and momentum distribution also agree with those of Bogolyubov theory at low density, despite the fact that it is based on ideas that are very different from those of Bogolyubov theory. In addition, another equation of the approach called the Big equation has been found to yield numerically accurate results for these observables over the entire range of densities for certain interaction potentials. This book is an introduction to Lieb's Simplified approach, and little background knowledge is assumed. We begin with a discussion of Bose gases and quantum statistical mechanics, and the notion of Bose-Einstein condensation, which is one of the main motivations for the approach. We then move on to an abridged bibliographical overview on known theorems and conjectures about Bose gases in the thermodynamic limit. Next, we introduce Lieb's Simplified approach, and its derivation from the many-body problem. We then give an overview of results, both analytical and numerical, on the predictions of the approach. We then conclude with a list of open problems.

Figures (4)

• Figure 1: The phase diagram of the ideal gas of Helium-4 atoms. The $y$-axis is the mass density.
• Figure 2: CHe21 The predictions of the energy per particle as a function of the density for the Simple Equation, Medium Equation, and Big Equation, compared to a Quantum Monte Carlo (QMC) simulation (computed by M.- Holzmann) and the Lee-Huang-Yang (LHY) formula, for $v=e^{-|x|}$.
• Figure 3: The predictions of the condensate fraction as a function of the density for the Simple Equation, Medium Equation, and Big Equation, compared to a Quantum Monte Carlo (QMC) simulation (computed by M.- Holzmann) and the Bogolyubov prediction (Bog), for $v=e^{-|x|}$.
• Figure 4: The predictions of the two-point correlation function for the Simple Equation, Medium Equation, and Big Equation, compared to a Quantum Monte Carlo (QMC) simulation (computed by M.- Holzmann), for $\rho=0.0001$ and $\rho=0.02$ and $v=16e^{-|x|}$.